Bat conjecture for finite length spectra of convex polygons

Let $n\geq 3$, and let $\Omega_1$ and $\Omega_2$ be convex $n$-gons. A primitive closed geodesic is a closed geodesic that is not a nontrivial iterate of a shorter closed geodesic; consider the lengths of such geodesics in each polygon. Bat conjecture. For each $n\geq 3$, there exists $N=N(n)$ such that, if the lengths of $N$ primitive closed geodesics of $\Omega_1$ and $\Omega_2$ coincide, then $\Omega_1$ and $\Omega_2$ are identical up to a rigid motion. This is motivated by the analogy with bats locating objects from finitely many sound-return times. The source gives no resolution.